Find the maximum area of a parallelogram drawn in the area enclosed by the curves y=4-x^2 & y=x^2+2x
We will use geogebra! Let's see if we can do this.
1) Type in f(x) = 4 - x^2
2) Type in g(x) = x^2 + 2x
3) Use the intersection tool on the two functions and the points A and B will be created.
4) Relabel B to C.
5) Create a slider called b set the interval to -2 to 1 with a step size .001. Type (b,f(b)). Point B will be created on f(x).
6) Use the line tool to create a line from A to B.
7) Use the parallel line tool to create a line that is parallel to AB and passes through C.
8) Get the intersection of this second line and g(x) using the intersection tool. Point D and E will be created. Hide E.
9) Create a line through BC.
10) Draw a line through D that is parallel to BC.
Notice that we now have a generic parallelogram drawn between the two curves.This is all we need!
11) You can hide the lines as best as you can. Create a polygon that uses A,B,C and D as its vertices.
12) Use the slider to get the maximum area. It is not difficult to get 6.75
13) You should have something close to the drawing shown below.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Thinking is cheating.
In what software are we going to do this?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.